CHAPTER
8
CALCULATION THEORY
.
Volume 2
CHAPTER 8 CALCULATION THEORY
Detailed in this chapter:
• the theories behind the program
• the equations and methods that
are use to perform the analyses.
CH8-1
CONTENTS
CHAPTER 8
CALCULATION THEORY.............................................................................................................3
8.1 GENERAL PILES................................................................................................................................................ 3
8.1.1 Vertical Analysis .....................................................................................................................................3
8.1.2
Lateral Analysis................................................................................................................................... 15
8.2 DRILLED SHAFT ANALYSES..................................................................................................................... 19
8.2.1
Vertical Analysis ................................................................................................................................. 19
8.2.2
Lateral Analysis................................................................................................................................... 24
8.3
SHALLOW FOOTING ANALYSES......................................................................................................... 25
8.3.1
Vertical Analysis ................................................................................................................................ 25
8.3.2
Capacity for Combined Loading..................................................................................................... 28
8.3.3
Settlement From Vertical Load......................................................................................................... 30
8.3.4
Rotation From Moment..................................................................................................................... 31
8.4
UPLIFT PLATE.............................................................................................................................................. 33
8.4.1
Shallow Mode ...................................................................................................................................... 33
8.4.2
Deep Mode........................................................................................................................................... 34
8.5 UPLIFT ANCHOR .......................................................................................................................................... 35
8.6 SOIL PARAMETERS AND CORRECTIONS ............................................................................................. 37
Appendix A Symbols and Notations
Appendix B Units Conversions
CH8-2
CHAPTER 8
CALCULATION THEORY
8.1 GENERAL PILES
8.1.1 Vertical Analysis
This program uses procedures described in the Foundations & Earth
Structures,Design Manual 7.02, published by Department of Navy, Naval
Facilities Engineering Command.
8.1.1.1 Downward (Compression) Load Capacity Calculation
Ultimate downward capacity can be determined by the following equations:
Qdw = Q tip + Qside
Where
Qdw = ultimate downward capacity
Q tip = ultimate tip resistance
Qside = ultimate side resistance
Ultimate tip resistance:
Qtip = Atip · qult = Atip · (Nq Sv + Nc)
Where
Atip = area of pile tip
qult= ultimate end bearing pressure
Sv = vertical stress in soil (overburden pressure)
Nq = bearing factor for cohesionless soils. It is a function of
friction shown in Table 8-1.
Nc = bearing factor for cohesive soils. It is a function of z/B
(depth/width) shown in Table 8-2.
Table 8-1. Bearing Capacity Factor, Nq
φ
Nq
Nq
(Internal friction)
(Displacement pile)
(No-Displacement pile)
26
10
5
28
15
8
30
21
10
31
24
12
32
29
14
33
35
17
CH8-3
34
42
21
35
50
25
36
62
30
37
77
38
38
86
43
39
120
60
40
145
72
Table 8-2. Bearing Capacity Factor, Nc
z/B
Nc
(Depth/Width)
0
6.3
1
7.8
2
8.4
3
8.8
4
9
>4
9
Ultimate side resistance:
Q side = Σ Sf P i ∆l = Σ (f0 + Ca) Pi ∆l
Where
Sf = side resistance
f0 = skin friction of cohesionless soil
Ca = adhesion of cohesive soil
P i = Perimeter of pile section
∆l = segment of pile
Skin friction of cohesionless soil:
f0 = Sh tan(d) = Kdown · Sv · tan(d)
Where
Sh
or S h = K down ⋅ S v
Sv
Sv = vertical stress in soil
K down =
Sh = horizontal stress in soil
Kdown = ratio of Sh/Sv which is defined in the table of Setup.
CH8-4
d = skin friction between soil and pile. It is a function of pile skin
materials. For steel pile, d = 20o-30o. For concrete pile, d = Kf
φ. Kf is friction factor ranging from 0.1 to 1. Kf can be defined
in the table of Setup.
Adhesion of cohesive soil:
Ca = Kc · Ka · C
Where C = shear strength of cohesive soil (cohesion)
Kc = adhesion factor ranging from 0.1 to 1, defined in the
table of Setup.
Ka = Adhesion ratio, Ca/C, which is a function of C shown
in Figure 8-1.
Adhasion Ratio Ka
1.5
1.25
All Piles
Ration Ka=Ca/C
1
Concrete Pile
0.75
0.5
0.25
0
0
1000
2000
3000
Cohesion C, Psf
Figure 8-1. Adhesion Ratio, Ka
CH8-5
4000
Limited Depth of side resistance and end bearing :
Experience and field evidence indicate that the side friction and end bearing
increase with vertical stress Sv up to a limiting depth of embedment. Beyond
this limiting depth (10D to 20D, D-pile width), there is very little increase in
side friction and end bearing. Penetration Ratio, PR, is used to define the
limiting depth. PR = 20 is commonly used for both side friction and end
bearing. The values can be changed on the Advanced page.
PRtip = (z/D)tip, Penetration Ratio for calculation of end bearing.
PR side = (z/D)side, Penetration Ratio for calculation of side resistance.
Where
z = depth
D = average pile width
The limitation of side friction and end bearing also can be expressed as
absolute value for both cases. The values can be changed in the table of
Advanced Page.
q_ limit, Limit of end bearing pressure.
Sf_ limit, Limit of sum of side friction and adhesion.
Allowable downward capacity can be determined by the following: equation:
Qallw _ d =
Qtip
FS tip
Where
+
Q side
FS side
Qtip = ultimate tip resistance
Qside = ultimate side resistance
FStip = factor of safety for tip resistance, defined in the
table of Advanced Page.
FSside = factor of safety for side resistance in downward
direction, defined in Advanced Page.
8.1.1.2 Zero Side Resistance
In some cases, a portion of the pile does not have contact with soils. For
example, soils have gaps, or the pile passes through an underground basement
or tunnel. Side resistance cannot be developed in this portion. Therefore the
concept of zero friction can be used. It includes both zero friction and zero
adhesion. Two zero-resistance zones can be input in the program.
CH8-6
8.1.1.3 Zero Tip Resistance
In special conditions, users do not want to include the tip resistance in pile
capacity. These conditions include peat or soft soils at pile tip. Or the pile tip
has a very sharp point. Users can include the depth of the pile tip in the zero
resistance zones. For example, if the pile tip is at a depth of 35 feet, users can
set a zero resistance zone from 35 to 36 feet. The tip resistance will be zero in
the calculation.
8.1.1.4 Negative Side Resistance
Piles installed through compressive soils can experience “downdrag” forces or
negative resistance along the shaft, which results from downward movement
(settlement) of adjacent soil. Negative resistance results primarily from
consolidation of soft deposits caused by dewatering or fill placement. The
downdrag force is the sum of negative friction and adhesion. It does not
include tip resistance. It only effects downward capacity, not uplift capacity.
Two zero- and two negative-resistance zones can be input in the program. If
the same zone is defined as both a zero-resistance and negative-resistance
zone, the program considers the zone as a zero-resistance area.
Downdrag Force from Negative Friction:
Q neg = Kneg ·Σ (Sf) Pi ∆l = Kneg ·Σ (f0 + Ca) Pi ∆l
Where
Q neg = Downdrag force from negative side friction
Kneg = Negative side friction factor. It ranges from 0 to 1
depending on the impact of settlement of the soil to the pile shaft.
Sf = side resistance
f0 = skin friction of cohesionless soil
Ca = adhesion of cohesive soil
Pi = Perimeter of pile section
∆l = segment of pile
8.1.1.5 Maximum Settlement Calculation at Ultimate Vertical Resistance
Based on Vesic’s recommendation (1977), the settlement at the top of the pile
consists of the following three components:
Settlement due to axial deformation of pile shaft, Xs
∆l
A' E
Qtip = tip ultimate resistance
X s = ∑(Qtip + Qside)
Where
Qside = side ultimate resistance
∆l = pile segment
A’ = effective pile cross sectional area
E = modulus of elasticity of the pile
CH8-7
The equation is different from what shown in DM-7. This equation uses numerical integration,
which is more accurate then the empirical equation in DM-7.
Settlement of pile point caused by load transmitted at the point, Xpp
X pp =
Where
C pQ t
Bq ult
Cp = empirical coefficient depending on soil type and method of
construction. It is defined in Table 8-3 below.
B = pile diameter
qult= ultimate end bearing pressure
Table 8-3. Typical Value of Cp for Settlement Analysis
Soil Type
Driven Piles
Drilled Piles
Sand
0.03
0.135
Clay
0.025
0.045
Silt
0.04
0.105
Settlement of pile point caused by load transmitted along the pile shaft, Xps
X ps =
Where
C sQs
Le q ult
Le = embedded depth
qult= ultimate end bearing pressure
Qs = side resistance
C s = (0.93 + 0.16
Where
z
)C
B p
z/B = depth / pile width
(Note: NAVY DM-7 has typo mistake in the equation)
Total settlement of a single pile, Xtotal
Xtotal = (Xs+Xpp+Xps)
CH8-8
8.1.1.6 Relationship Between Settlement and Vertical Load
Vertical load and settlement relation can be developed from t-z (side load vs
shift movement) and q-w (bearing load vs base settlement) curves. The t-z
curve represents the relation between side resistance and relative movement
within soil and shaft. The t-z curve can vary at different depth and in different
soils. The q-w curve represents the relation between tip resistance and base
movement of the shaft.
t-z and q-w Relation
Generally, t-z and q-w relations require a considerable amount of geotechnical
data from field and laboratory testings, which are not always available for
engineers. AllPile uses the following procedures to determine the amount of
settlement:
1. First, calculate ultimate side resistance and ultimate tip resistance of
shaft using the methods introduced in 8.1.1.5.
2. Find relationships between settlement and load transfer ratios
(developed resistance against ultimate resistance) using the
corresponding charts in Fig 8-2 – 8-5
3. Integrate both side and tip resistances, as well as elastic
compression of shaft body, to obtain total vertical resistance as a
function of settlement.
4. From the relationships between settlement and load transfer ratios,
we can develop t-z and q-w curve.
Typical settlement against load
transfer ratios are shown in Figures 82 through 8-5 proposed by Reese and
O’Neal (1988). Figure 8-2 and 8-3
represent the side load transfer ratio
for cohesive soils and cohesionless
soils/gravel respectively. Figure 8-4
and 8-5 represent the end bearing load
transfer ratio for cohesive soils and
cohesionless soils respectively.
Figure 8-2.
Normalized load transfer
relations for side
resistance in cohesive soil
(Reese and O'Neill, 1989)
CH8-9
Figure 8-3 Normalized load transfer relations for side resistance in cohesionless
soil (Reese and O'Neill, 1989)
Figure 8-4. Normalized load transfer relations for base resistance in
cohesive soil (Reese and O'Neill, 1989)
CH8-10
Figure 8-5. Normalized load transfer relations for base resistance in
cohesionless soil (Reese and O'Neill, 1989)
Two Options for Settlement Analysis
In Advanced Page, AllPile provides two options for developing load-settlement
relation.
Option 1:
The load transfer ratio is based on diameter of shaft (Ds) or base
diameter of shaft (Db) if it is different from the former, i.e. shafts with
bell. This option is recommended for larger-size shafts.
Option 2:
The load transfer ratio is based on the calculated settlement from
Vesic's method as described in Section 8.1.1.5. This option yields a
closer match between settlement calculation of Vesic’s method. It is
recommended for smaller diameter piles.
CH8-11
Total, Side and Tip Resistance vs. Settlement
Figure 8-6 shows the vertical load is distributed in to side resistance and tip
resistance. The chart from results of program shows that side resistance
develops at small settlement, while tip resistance develops at large settlement.
The ultimate value of the two cannot simply be added together. That is why
tip resistance requires large Factor of Safety to get allowable capacity.
Figure 8-6. Total, Side and Tip Resistance vs. Settlement
Capacity at Allowable Settlement
AllPile provides two methods to determine Qallow. One is defined by Factor of
Saftey presented in Section 8.1.1.1. The other method is defined by allowable
settlement.
Calculate Qallow based on allowable settlement. Depending on the amount of
allowable settlement Xallow, then back-calculate Qallow based on the relationship
between Xallow and load. Xallow can be defined on Advanced Page.
8.1.1.7 Uplift Load Capacity Calculation
Ultimate uplift capacity can be determined by the following equations:
Qup = Qw + Qside
CH8-12
Where
Qw = weight of pile
Qside = ultimate side resistance
Qw = ΣWi ∆l
Where
Wi = weight of pile section in unit length
∆l = segment of pile
Q side = Σ Sf P i·∆l = Σ (f0 + Ca) Pi ∆l
Where
Sf = side resistance
f0 = skin friction of cohesionless soil
Ca = adhesion of cohesive soil
∆l = segment of pile
P i = Perimeter of pile section
f0 = Kup ·Sv tand
K up =
Where
Sh
Sv
or
S h = K up ⋅ S v
Sv = vertical stress in soil
Sh = horizontal stress in soil
Kup = ratio of Sh/Sv which is defined in the table of Setup
d = skin friction between soil and pile. It is function of pile side
materials. For steel pile, d = 20o-30o. For concrete pile, d = Kf? . Kf
is friction factor ranging from 0.1 to 1. Kf can be defined in in the
table of Setup.
Ca = Kc · Ka · C
Where C = shear strength of cohesive soil (cohesion)
Kc = adhesion factor ranging from 0.1 to 1, defined in the table of
Setup.
Ka = Adhesion ratio, Ca/C, which is a function of C shown in Figure
8-1.
Allowable Uplift Capacity can be determined by following equations:
Qallw_ U =
Qw
Q
+ side
FS _ w FS _ up
CH8-13
Where
Qw = weight of pile
Qside = ultimate side uplift resistance
FS_w = factor of safety for pile weight, defined in the table of
Advanced Page.
FS_up = factor of safety for side resistance for uplift, defined in the
table of Advanced Page.
8.1.1.8 Batter Shaft Capacities Calculation
The capacities of batter is from vertical capacities then adjusted by its batter
angle:
Q batter = cos α · Q vertical
Where
α = Batter angle of shaft
Q = vertical capacities including downward and uplift
8.1.1.9 Group Vertical Analysis
In most cases, piles are used in groups as shown in Figure 8-7, to transmit the
load to each pile. A pile cap is constructed over group piles. The analysis can
be divided into four steps.
Figure 8-7 Group Pile for Vertical Analysis
Step 1. Calculate Capacity of Individual Pile, Q single
Qsingle can be calculated using the methods mentioned in above sections.
Qsingle includes side resistance and tip resistance.
Step 2. Calculate Capacity of a Pile Block, Qblock
Qblock is calculated using single pile method including side and tip resistance.
The block has the following dimensions:
Bx = (nx -1) · Sx + D
CH8-14
By = (ny -1) · Sy + D
L is the same as the length of each individual pile
Step 3. Calculate the Group Efficiency
η=
Qblock
nQsin gle
Where
n = total number of pile. n = nx · ny
Qsingle = capacity of individual pile
Qblock = capacity of block pile
η = group efficiency
Step 4. Determine the Capacity of Group Pile, Qgroup
If η = 1, then Qgroup = n · Qsingle
If η < 1, then Qgroup = Qblock
8.1.1.10
Settlement Analysis for Group Pile
Suggested by Vesic (1969), the settlement for group pile can be estimated
X group = X sin gle ⋅
B'
D
based on settlement of a single pile (DM7-7.2-209):
Where
B' = smallest dimension between Bx and By (see Step 2 above)
D = diameter of a single pile
8.1.2
Lateral Analysis
AllPile directly uses COM624P calculation methods for lateral analysis. For
details on COM624P, please refer to the FHWA publications, FHWA-SA-91048, COM624P – Laterally Loaded Pile Program for the Microcomputer,
Version 2.0, by Wang and Reese (1993). In that publication, Part I provides a
User’s Guide, Part II presents the theoretical background on which the
program is based, and Part III deals with system maintenance. The
appendices include useful guidelines for integrating COM624P analyses into
the overall design process for laterally loaded deep foundations.
CH8-15
8.1.2.1 Lateral Deflection Calculation
Here is brief introduction to the program. COM624P uses the four nonlinear
differential equations to perform the lateral analysis. They are:
d 4Y
d 2Y
EI 4 + Q 2 − R − Pq = 0
dZ
dZ
Where
(1)
Q = axial compression load on the pile
Y = lateral deflection of pile at depth of Z
Z = depth from top of pile
R = soil reaction per unit length
E = modules of elasticity of pile
I = moment of inertia of the pile
P q = distributed load along the length of pile
EI (
d 3Y
dY
) + Q( ) = P
dZ
dZ
(2)
EI (
d 2Y
)=M
dZ 2
(3)
Where
P = shear in the pile
Where
M = bending moment of the pile
dY
= St
dZ
( 4)
Where St = slope of the elastic curve defined by the axis of pile
The COM624P program solves the nonlinear differential equations
representing the behavior of the pile -soil system to lateral (shear and moment)
loading conditions in a finite difference formulation using Reese’s p-y method
of analysis. For each set of applied boundary loads the program performs an
iterative solution, which satisfies static equilibrium and achieves and
acceptable compatibility between force and deflection (p and y) in every
element.
Graphical presentations versus depth include the computed deflection, slope,
moment, and shear in the pile, and soil reaction forces similar to those
illustrated in Figure 8-8.
CH8-16
Figure 8-8 Graphical Presentation of AllPile Results
Figure 8-9 Group Pile for Horizontal Analysis
CH8-17
8.1.2.2
Group Lateral Analysis
Due to the group effect, the lateral capacity of individual piles can not be fully
developed. Deduction factors are applied to the soil reaction, then lateral analysis
is performed for individual piles.
Step 1. Calculate Deduction Factor Rside and Rfront
Assuming the lateral load P is in X direction. Please note that Rfront is not the
same as Rside.
Table 8-4. Deduction Factor Rfront
Sx
Rfront
>8D
1
8D
1
6D
0.8
4D
0.5
3D
0.4
<3D
0.4
Table 8-5. Deduction Factor Rside
Sy
Rside
>3D
1
3D
1
2D
0.6
1D
0.3
<1D
0.3
Note: D = pile diameter
Reference: FHWA HI 97-013
Step 2. Reducing Soil Lateral Resistance by Applying the Deduction Factors from Step 1
Combine the reduction factors and apply them to p-y curve for lateral analysis.
Step 3. Calculate P-yt (the Lateral Capacity and Deflection) of Each Individual Piles
Calculate the lateral capacity of each pile and get P-Yt curve for all piles.
Step 4. Get Lateral Capacity of group piles
After P-Yt curve for each pile is constructed. The Pgroup is the sum of Psingle for
individual piles at the same deflection under one pile cap.
P group = ΣP single
y group = y single
CH8-18
8.2 DRILLED SHAFT ANALYSES
Drilled shaft is normally used in deep
foundation to transfer vertical load
through weak soils to stronger soils or
rocks at depth. Since it is often used to
carry a relatively large vertical load
over a good depth of soils, typical
diameter of drilled shaft ranges from 4
ft (1.2 m) to 20 ft (6 m). In most cases,
the aspect ratio of a drilled shaft, or its
length divided by its diameter, should
not exceed 30.
This program uses procedures
described in the Drilled Shafts:
Construction Procedures and Design
Methods (FHWA-IF-99-025) published
by FHWA in August 1999.
Figure 8-10. Drilled Shaft
8.2.1
Vertical Analysis
8.2.1.1 Downward (Compression) Load Capacity Calculation
Ultimate downward capacity can be determined by the following equations:
Qdw = Q tip + Qside
Where
Qdw = ultimate downward capacity
Q tip = ultimate tip resistance
Qside = ultimate side resistance
Ultimate tip resistance ( Qtip ):
Base in cohesive soils [Su ≤ 0.25 MPa (5,200psf)]
Qtip = q ult Ab
Where
qult = ultimate bearing pressure
Ab = base area
If Z (depth of base) ≥ 3Db (diameter of base):
qult = 9 Su
or qult = Nc* ·Su
[If Su ≥ 96kPa (1tsf)]
[If Su < 96kPa (1tsf)]
CH8-19
Where
Su = undrained shear strength below base
Nc* = modified bearing capacity factor for cohesive soils. It
can be assumed to be a function of Su in UU triaxial
compression as shown in Table 8-6.
Table 8-6. Modified Bearing Capacity Factor, Nc*
Su
Nc*
(Undrained Shear Strength)
(Bearing Capacity Factor)
24kPa (500psf)
6.5
48kPa (1000psf)
8.0
96kPa (2000psf)
9.0
If Z (depth of base) ≥ 3Db (diameter of base):
q ult =
2 
Z 
+ 1 +
 N c * ⋅S u
3  6 Db 
Base in cohesionless soils (NSPT ≤ 50)
In English unit:
qult (kPa) = 57.5 · NSPT
In Metric unit:
qult (tsf) = 0.6 · NSPT
Where
NSPT = blow count per 0.3m or 1ft of penetration in the Standard
Penetration Test
Base in rocks [0.25MPa (2.5tsf) < Su < 2.5MPa (25tsf)]
If embedment in rock ≥ 1.5Db (diameter of base):
qult = 5 · Su = 2.5 · qu
Where
qu = unconfined compressive strength below base
*Attention: The equation above are developed for drilled shafts embedded at least 1.5Db
(diameter of base) in good quality bedrock with RQD close to 100%. If rock is jointed or
fractured, please consult geotechnical engineer for correct procedures to calculate tip
resistance.
CH8-20
Ultimate side resistance ( Qside ):
Q side = Σ f0 ·P i ∆l ·
Where
f0 = skin friction
∆l = segment of pile
P i = perimeter of pile
Shaft in cohesive soils [Su ≤ 0.25 MPa (5,200psf)]
f0 = α · Sv
α = 0.55
(for Su / Pa ≤ 1.5)
 Su

α = 0.55 − 0.1 − 1.5 
 Pa

(for 1.5 ≤ Su / Pa ≤ 2.5)
Where α = shear strength reduction factor
P a = atmospheric pressure = 101kPa or 2.12ksf
Shaft in cohesionless soils (NSPT ≤ 50)
f0 = β · Sv'
In sand:
β = 1.5 - 0.245 [Z(m)]0.5
[If NSPT ≥ 15]
or β = NSPT /1.5 · {1.5 - 0.245 [Z(m)]0.5}
[If NSPT ≥ 15]
In gravelly sand or gravel:
β = 2.0 - 0.15 [Z(m)]0.5
[If NSPT ≥ 15]
or β = NSPT /1.5 · {1.5 - 0.245 [Z(m)]0.5}
[If NSPT ≥ 15]
Where β = empirical factor which varies with depth
Sv = effective vertical stress at depth Z
Z = depth where side resistance is calculated
Attention:
- Z must be converted to meter before calculating β.
- Range of β: 0.25 ≤ β ≤ 1.2
Shaft in rocks [0.25MPa (2.5tsf) < Su < 2.5MPa (25tsf)]
 qu 
f0 = 0.65 ⋅ Pa 
 Pa 
0 .5
CH8-21
Where
qu = unconfined compressive strength at depth where side
resistance is calculated
P a = atmospheric pressure = 101kPa or 2.12ksf
8.2.1.2 Uplift Load Capacity Calculation
Ultimate uplift capacity can be determined by the following equations:
Qup = Qw + Q'side + Q'b
Where
Qw = weight of pile
Q'side = ultimate side resistance against uplift
Q'b = ultimate bell resistance against uplift (Q'b is only calculated
for belled shafts in cohesive soils)
Qw = Σ Wi ·∆l
Where
Wi = weight of pile section in unit length
∆l = segment of pile
Q' side = Σ k· Qside
Where
k = coefficient of uplift resistance
k=1
(for cohesive soils)
k = 0.75
(for cohesionless soils)
k = 0.7
(for rocks)
Qside = ultimate side resistance in compression in Section 8.2.1.1
If a belled drilled shaft is used
Q'b = Nu · Su· A'b
Where
(for cohesive soils only)
Nu = bearing capacity factor for uplift
= 3.5 Z/Db or 9 (whichever is smaller)
Z = depth of drilled shaft
Db = diameter of base/bell
Su = undrained shear strength
A'b = area of bell base minus area of shaft body ("Donut" area)
Attention: Belled shaft is not recommended for cohesionless soil
and is too difficult to be constructed in rock layer. Therefore, Q'b
will not be considered in those two types of earth material.
CH8-22
Shaded area = A'b
Bell
Shaft body
Figure 8-11 Top View of Donut Area
8.2.1.3 Exclusion Zones
According to the SHAFT manual, the exclusion zones do not contribute side
resistance for drilled shaft as shown in Figure 8-10.
Exclusion zones in the calculation of Downward Capacity :
•
For straight shafts: Top 5' and bottom one diameter of shaft
•
For belled shafts: Top 5' belled section and one diameter of stem (Ds)
Exclusion zones in the calculation of Uplift Capacity :
•
For straight shafts: Top 5'
•
For belled shafts: Top 5', entire belled section and two diameter of stem
(Ds) calculated from top of belled section
8.2.1.3 Group Vertical Analysis
Min. width of group
Bg =
In most cases, shafts are used in group as shown in Figure 8-12, to transfer
the load to each shaft. A cap is constructed over group shafts. The analysis
can be divided into four steps.
Ds = Diameter of shaft
Figure 8-12 Group Shaft for Vertical Analysis
CH8-23
Step 1. Calculate Capacity of Individual Pile, Q single
Qsingle can be calculated using the methods mentioned in above sections.
Qsingle includes side resistance and tip resistance.
Step 2. Calculate Minimum (Shortest) Dimension of Shaft Block, B g
Bg = (Nx-1) · Sg + Ds
Where
Nx = number of shafts on the short side of the group
Sg = shaft spacing
Ds = diameter of drilled shaft
Step 3. Calculate the Group Efficiency, ?
η=
Where
Bg
Ds
Bg = minimum width of shaft group
Ds = diameter of drilled shaft
Step 4. Determine the Capacity of Group Pile, Qgroup
Qgroup = η · Qsingle
8.2.2
Lateral Analysis
Lateral analysis for drilled shafts at single or group conditions are identical to
that for drilled or driven piles. User can refer to Section 8.1.2 for the theories
and the calculation procedures used in lateral analysis.
CH8-24
8.3 SHALLOW FOOTING ANALYSES
Figure 8-13. Shallow Footing
8.3.1
8.3.1.1
Shallow foundation is designed to transfer
vertical load to relatively shallow depth through
end bearing. Typical shallow foundations
include spread footings, strip footings, and met
footings. The bearing capacity of shallow
foundation is influenced by a number of factors,
which will be covered in the next section. On
the other hand, shallow foundation is often
subject to lateral loading or eccentricity. The
stability of shallow foundation against
eccentricity is controlled primarily by its ability
to withstand overturning. AllPile uses
procedures and recommendations given in
Principles of Foundation Engineering,
Brooks/Cole Engineering Division, Braja M.
Das., 1984, as the primary references for
shallow foundation analyses.
Vertical Analysis
Vertical (Compression) Load Capacity Calculation
Ultimate downward capacity (qult) can be determined by the following
equation:
qult = c N c s c d c i c g c + q N q sq d q i q g q + 0.5 γ D N γ s γ d γ i γ g γ
Where
c = cohesion
q = effective stress of soil at foundation base
γ = unit weight of soil
D = width or diameter of foundation
N = bearing capacity factors
s = shape factors
d = depth factors
i = load inclination factors
g = ground inclination factors
a.) Bearing Capacity Factors (N):
N c = (N q − 1)cot φ
φ

N q = tan 2  45 + e π tan φ
2

CH8-25
N γ = 2(N q + 1) tan φ
b.) Shape Factor (s):
 B  N q
sc = 1 +  
 D  N c




B
s q = 1 +   tan φ
D
 B
sγ = 1 − 0.4 
 D
Where, B = Length of footing
D = Width of footing
c.) Depth Factor (d):
For shallow foundations, in which embedment to footing width ratio
(L/D) ≤ 1:
L
d c = 1 + 0 .4  
 D
d q = 1 + 2 tan φ (1 − sin φ )
2
L
D
dγ = 1
For deeper foundations, in which L/D > 1:
L
d c = 1 + 0.4 tan −1  
 D
 L
d q = 1 + 2 tan φ (1 − sin φ )2 tan −1  
D
dγ = 1
Where, tan-1 (L/D) is in radius
Al
d.) Load Inclination Factor (i):
A 

ic = iq =  1 − l 
 90° 
CH8-26

A
iγ = 1 − l 
φ 

Where, Al is inclination of load in degree. Al = tan-1 (P/Q) is in radius.
P = Shear Load, Q = Vertical Load
e.) Ground Inclination Factors (g) (Reference: Foundation Design
Principles & Practices, Donald P. Conduto, p.176):
A 

g c = 1 − s 
 147 
g c = g γ = (1 − 0.5 tan As )
5
Where, As is angle of slope in degree.
f.) Battered Footing Reduction Factors (kbat ):
As
Ab
k bat = cos ( Ab )
Where, Ab is angle of battered footing against
vertical axis in degree.
*Attention: Unlike other factors, kbat is not applied to the equation of
ultimate downward capacity (qu) directly. It will be put into the
calculation when the total ultimate downward capacity (Qu) is
calculated. Detail about Qu is given below.
Total ultimate downward capacity (Qu) represents the total bearing capacity
against compression over the area of footing base. It can be determined by
the following equations:
Net Ultimate Bearing Capacity (qnet ):
qnet = qu - q
Where, qu = ultimate bearing capacity
q = overburden soil pressure
Total Ultimate Bearing Capacity (Qult):
Qult = (qnet x kbat ) A
Where, kbat = battered footing reduction factor
CH8-27
A = base area of footing
Allowable downward capacity (Qallow) can be calculated by the following
equation:
Q allow =
8.3.2
8.3.2.1
Q ult
F .S .
Capacity for Combined Loading
Vertical (Compression) Load (Q) Only
If there is only a vertical load, Q, without any lateral loading, i.e. shear loads
and bending moment, the Factor of Safety of the shallow foundation can be
calculated using the equation below:
F .S. =
Q ult
Q.
Where, Qult = ultimate bearing capacity
Q = total vertical load
Users can also check the ratio between Q and the allowable bearing capacity,
Qallow, to see if the shallow foundation is considered stable. If Q > Qallow, the
foundation is insufficient.
8.3.2.2
Vertical Load With Moment (Q + M)
Typical lateral loads on the foundation include bending moment (M) and shear
load (P) as illustrated in the next diagram. In this section, we will study the
procedures used to determine footing capacity against the combination of
vertical load and bending moment (Q+M).
Eccentricity (e) will be generated by the moment and vertical load (see Figure
8-12):
e=
M
Q
CH8-28
a.) If e ≤ D/6, the pressure on the foundation can be determined by:
Q
6M
+ 2
DB D B
Q
6M
=
− 2
DB D B
q max =
q min
Where, D = width of foundation base in lateral load direction
B = length of foundation base in the other direction
Reaction pressure at the base of the foundation distributed in a trapezoid
pattern across the full width (D) of the foundation.
b.) If e > D/6, then:
q max =
4Q
4 B( D − 2e)
q min = 0
Reaction pressure at the base of the foundation is distributed in a triangular
pattern across the effective width (D') of the foundation.
D' = D − 2 e
Due to the distribution of reaction pressure, a new ultimate bearing capacity
called, qult', has to be recalculated using the same procedures as mentioned
in Section 8.3.1.1, but based on D' instead of D.
To calculate the Factor of Safety:
F .S . =
q ult '
q max
or
F .S. =
Q ult '
Q
Where, Qult' = qult' ⋅ D' ⋅ B
8.3.2.3
Vertical Load With Shear Load (Q + P)
The shear load (P) has two impacts to the shallow foundation calculation:
CH8-29
1. It generates load inclination − Al = tan-1 (P/Q) − which affects vertical
bearing capacity calculation (see Section 8.3.1.1).
2. Footing base sliding calc ulation becomes necessary. The sliding
resistance (P f) can be calculated by the following equation:
Pf = k f (W + Q)
Where, kf = base friction factor for cast-in-place foundation
kf is close to tanφ (φ = angle of internal friction)
kf = 0.3-0.8 is recommended
W = weight of footing and the soil above
Factor of Safety against sliding can be calculated by:
F .S. =
8.3.3
Pf
P
Settlement From Vertical Load
If only vertical load is applied to the shallow foundation, the elastic settlement
(X0) of the footing can be calculated using the equation below:
X0 =
Dq 0
(1 − µ 2 )α
Es
Where, q0 = pressure under working load
µ = poisson ratio;
Q
q
or
q0 = u
Area
F .S .
= 0.3 is recommended for general soil conditions
q0 =
Es = Young's modulus
= 766NSPT
= 375C
for cohesionless soils
for cohesive soils
Where, NSPT = blow count over 12" of soil
C = undrained cohesion of soil
α = settlement factor for flexible foundation, which is
a function of D/B (footing shape ratio)
[Note 1] X0 is the elastic settlement at the center of a footing. If there is soft
clay underneath the footing, consolidation settlement, which is time-dependent,
CH8-30
should be considered. AllPile does not include calculation of consolidation
settlement as it is not within the scope of the program.
[Note 2] AllPile assumes that a hard layer of soil, i.e. rock or intermediate
geomaterials (IGMs) is in great depth from the base of footing. If Ha, the
distance between footing base and hard soil, is over 4 times of footing width
(D), the actual elastic settlement would not change considerably.
If Ha is less than 4D, the elastic settlement can be calculated based on the
following equation:
X 0 '= X 0
Ha
4D
( H a < 4)
Where, X0' = actual elastic settlement when Ha < 4D
X0 = elastic settlement based on Ha > 4D
Ha = distance between bottom of footing and hard soil
If user does not define Ha, AllPile will automatically search for the closest
hard soil stratum with NSPT ≥ 50 based on user's input in the Soil Property
page.
8.3.4
Rotation From Moment
The maximum settlement and rotation for a footing under both vertical and
lateral loads can be determined by the following procedures:
1. Calculate eccentricity and effective width (D') based on section 8.3.2.2.
2. Determine the ultimate capacity (Q'ult) under moment and vertical load
from Section 8.3.2.2
3. Determine the Factor of Safety under both moment and vertical load
F .S .1 =
Q ult
Q
4. Calculate the ultimate capacity under vertical load only (see Section
8.3.2.1)
Qult (v)
5. Get Qallow (v) under vertical load only
Q allow (v ) =
Qult ( v)
F .S .1
CH8-31
6. Calculate X0 under Qallow (v) based on Section 8.3.3
7. Determine the maximum settlement and rotation using the equations
below:
2
3

e
e
e 
X max = X 0 1 + 2.31  − 22.61  + 31.54  
 D
 D
 D  

2
3

e
e
e 
X e = X 0 1 − 1.63  − 2.63  + 5.83  
D
 D
 D  




X
− Xe 
Rt ( Rotation ) = sin −1  max

 D −e 
 2

Where, Xmax = settlement at edges of footing
Xe = settlement under point of vertical load
(vertical load may not apply to center of footing)
e = eccentricity
Rt = Rotation of Footing
*Note: These equations are only valid if e/D ≤ 0.4
CH8-32
8.4
UPLIFT PLATE
Uplift plate is commonly used as a ground anchor to stabilize structure that is
under shear load or moment. Due to its characteristics, uplift plate only
provides uplift resistance against pulling, and has no bearing capacity. Uplift
plate calculation can be divided into two modes:
•
Shallow mode if L (= embedment) ≤ Lcr
•
Deep mode if L > Lcr
Where Lcr = critical depth in uplift resistance calculation
For cohesionless soils, Lcr is defined in Figure 8-14; For
cohesive soils, Lcr can be determined using the following
equation:
Lcr = D (0.107 Cu + 2.5)
Lcr ≤ 7D
Where Cu = undrained cohesion in kPa
Q
Q
L
Lcr
L
Shallow Mode
L-Lcr
Deep Mode
Figure 8-14 Critical Depth of Uplift Plates
8.4.1
Shallow Mode
For Cohesionless Soils
Uplift capacity (Quplift ) can be determined by the following equation:
Quplift = Bq ⋅ A ⋅ γL + W
CH8-33
Where, A = area of plate
W = weight of plate
Bq = breakout factor
 L 
L
Bq = 2  K u ' tan φ m  + 1 + 1
 D
  D 
Where, D = width of plate
Ku' = uplift factor, equal to 0.9 in general
φ = internal angle of friction of soil
m = shape factor coefficient, a function of φ
and is defined in figure 8-15
For Cohesive Soils
Uplift capacity (Quplift ) can be determined by the following equation:
Quplift = (C u ⋅ Bc + γL ) A + W
Where, Bc = breakout factor, can be determined using
the Figure 7-14 on page 78
Cu = undrained cohesion in kPa
A = area of plate
W = weight of plate
8.4.2
Deep Mode
For Cohesionless Soils
Uplift capacity (Quplift ) in deep mode can be determined by the following
equation:
Quplift = Q ' plate +Q ' skin
Where, Q'plate = uplift capacity calculated in shallow
mode
Q'side = side resistance developed in the portion
of (L - Lcr)
CH8-34
Critical Depth (Lcr)
14
12
10
4
3
2
y = 2.596E-06x - 1.168E-04x + 3.505E-03x + 1.907E-02x + 1.229E+00
Lcr/D
8
Figure 8-15 The
relationship between
critical depth (Lcr)
and friction angle of
soil (Phi)
6
4
2
0
0
5
10
15
20
25
30
35
40
45
50
Phi (deg)
Shape Factor Coefficient (m)
0.8
0.7
4
3
2
y = 5.370E-07x - 6.389E-05x + 3.218E-03x - 6.363E-02x + 4.618E-01
0.6
m
0.5
Figure 8-16 The
relationship between
shape factor coefficient
(m) and friction angle
of soil (Phi)
0.4
0.3
0.2
0.1
0
20
25
30
35
40
45
50
Phi (deg)
Breakout Factor Bc
1.2
1
4
3
2
y = 0.6818x - 1.493x + 0.085x + 1.7209x + 0.0034
Bc/9
0.8
Figure 8-17 The
relationship between
breakout factor (Bc)
and the ratio of
embedment against
critical depth
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
L/D over L cr/ D
CH8-35
1
8.5 UPLIFT ANCHOR
Uplift anchor has the same function of an uplift plate, though it uses a
completely different mechanism. Unlike uplift plate, which uses bearing
capacity generated from its base plate against the soil mass on top of the plate
to resist uplift force, uplift anchor generates the majority of its uplift resistance
through adhesion and friction from its grouted section. An uplift plate can be
divided into two portions.
•
The top section, formed by uncovered steel bar which extends from
the ground surface to the top of the grout, is typically called Free
Length (Lf). Friction developed in this section is neglected.
•
The bottom section is the grouted portion of the uplift anchor with a
diameter of D. The total side resistance generated in this section is
based on the adhesion of the grout and the bonded length (Lb).
The amount of adhesion is depended on grout pressure. The higher the grout
pressure, the higher the adhesion that can be achieved from the bonded length.
Post-grout also helps to generate higher adhesion.
Q uplift = πD ⋅ L b ⋅ C a
Where, Lb = bonded length
Ca = adhesion input by user
Q
STEEL BAR
Lf
GROUT
Lb
D
Figure 8.18 Uplift anchor
CH8-36
8.6 SOIL PARAMETERS AND CORRELATIONs
There are a number of references in the industry that present the correlations
between soil parameters. The soil parameters function is useful if users only
have a few parameters availa ble and want to estimate the others to complete
the calculation. However, one should bear in mind that these correlations are
from various sources, statistical results of different soil types under different
conditions. The actual value may be different from the estimate given by the
correlation. Users should make their own judgement based on local experience
and local soil conditions and adjust the value accordingly.
Following are the references used to form the soil correlation in the program:
Table 8-7. General Soil Parameters for Sand
Compactness
SPT*
Relative Density
Friction
Unit Weight
Moist
Submerged
Very Loose
Loose
Medium
Dense
Very Dense
Dr
φ
Unit
-%
Deg
0-4
0-15
<28
4-10
15-35
28-30
10-30
35-65
30-36
30-50
65-85
36-41
>50
85-100
>42
γ
γ
pcf
pcf
<100
<60
95-125
55-65
110-130
60-70
110-140
65-85
>130
>75
Symbol
N SPT
*SPT -- Standard Penetration Test
Reference: Steel Sheet Piling Design Manual, USS, 1975, p.12
Table 8-8. General Soil Parameters for Clay
Consistency
SPT
UCS*
Shear Strength
Unit Weight
Saturated
Very Soft
Soft
Medium
Stiff
Very Stiff
Hard
Symbol
NSPT
Unit
--
0-2
2-4
4-8
8-16
16-32
>32
qu
Cu
pcf
psf
0-500
0-250
500-1000
250-500
1000-2000
500-1000
2000-4000
1000-2000
4000-8000
2000-4000
>8000
>4000
γ
pcf
<100
100-120
100-130
120-130
120-140
>130
*UCS -- Unconfined Compressive Strength
Reference: Steel Sheet Piling Design Manual, USS, 1975, p.12
CH8-37
k and e50:
There are two parameters that are particularly important for lateral pile analysis—
Modulus of Subgrade Reaction (k) and Soil Strain (E50). Modulus of subgrade
reaction is used in the equation Es = k x in COM624 analysis, where Es is the secant
modulus on a p-y curve and x is the depth below ground surface. The value of k
describes the increase in Es with depth. Please note that the k-value is not the same
as the coefficient of vertical subgrade reaction used to calculate elastic settlements of
shallow foundations. It is also different from the coefficient of lateral subgrade
reaction used in elastic pile analysis. (For more detail and example, please refer to
NAVY DM7, 2-235. COM624 uses nonlinear differential analysis.) On the other
hand, the soil strain E50 parameter is only applicable for clay soil and is obtained by
either lab testing or by correlation. The input value E50 represents the axial strain at
which 50% of the undrained shear strength is developed in a compression test. The
following two tables demonstrate the correlation of k and E50 with other soil
parameters for different soil type:
Table 8-9. Modulus of Subgrade Reaction (k) vs NSPT for Sand
Compactness
Symbol
NSPT
Unit
--
(Dry)
k
(Saturated)
k
kN/m
pci
3
kN/m
pci
SPT
MSR*
3
Loose
Medium
Dense
4-10
10-30
30-50
6790
25
5430
20
24430
90
16300
60
61000
225
33900
125
*MSR -- Modulus of Subgrade Reaction
Reference: Handbook on Design of Piles and Drilled Shafts Under lateral Load,
US Department of Transportation, 1984, p.64
Table 8-10. Modulus of Subgrade Reaction (k) and Soil Strain (E50)
vs NSPT for Clay
Consistency
Soft
Medium
Stiff
Very Stiff
Hard
Symbol
NSPT
Unit
--
2-4
4-8
8-16
16-32
>32
Shear Strength
Cu
kPa
psf
12-24
250-500
24-48
500-1000
48-96
1000-2000
96-192
2000-4000
192-383
>4000
MSR*
Static Loading
k
Cyclic Loading
k
kN/m
pci
3
kN/m
pci
%
8140
30
--2
27150
100
--1
136000
500
54300
200
0.7
271000
1000
108500
400
0.5
543000
2000
217000
800
0.4
SPT
Soil Strain
E50
3
Reference: Lateral Load Piles, Lymon C. Reese, p.97
CH8-38
APPENDIX A SYMBOLS AND NOTATIONS
Symbol
Description
English
Sv
Vertical stress in soil (overburden pressure)
ksf
kN/m2
Sh
Horizontal stress in soil
ksf
kN/m2
qult
Ultimate end bearing
ksf
kN/m2
Sf = f0 +Ca
Side resistance (ultimate), combination of skin friction
and adhesion
ksf
kN/m2
f0
Skin friction from friction of soils (ultimate)
ksf
kN/m2
Ca
Adhesion from cohesion of soils (ultimate)
ksf
kN/m2
FS_ work
Factor of safety at working load condition
--
--
FS_ side
FS for side resistance in downward calculation
--
--
FS_ up
FS for side resistance in uplift calculation
--
--
FS_ tip
FS for tip resistance in downward calculation
--
--
FS_ w
FS for weight of pile in uplift calculation
--
--
Qtip
Vertical tip resistance
kip
kN
Qup
Uplift ultimate capacity
kip
kN
Qdw
Downward (compression) ultimate capacity
kip
kN
Qneg
Load from negative friction
kip
kN
Qwork
Vertical work load or design load applied to pile
kip
kN
Qallw_u
Allowable uplift capacity
kip
kN
Qallw_d
Allowable downward capacity
kip
kN
Qgroup
Vertical capacity of group pile
kip
kN
Qsingle
Vertical capacity of single pile
kip
kN
Qplate
Vertical uplift capacity of plate or bell
kip
kN
dz or dl
Pile segment
ft
m
Kbat
Factor for battered pile
--
--
Rt
Base rotation
degree
degree
R,Rside or Rfront
Group reduction factor
--
--
yt or y
Lateral deflection
in
cm
x
Vertical settlement
in
cm
Lcr
Critical depth in uplift analysis
ft
m
CH8-39
Metric
APPENDIX B UNITS CONVERSIONS
English to Metric
Metric to English
1 ft = 0.3048 m
1 m = 3.281 ft
1 in = 2.54 cm = 25.4 mm
1 cm = 0.3937 in
1 lb = 4.448 N
1 mm = 0.03937 in
1 kip = 4.448 kN
1 N = 0.2248 lb
1lb/ft2 = 47.88 N/m2
1 kN = 224.8 lb = 0.2248 kip
1 kip/ft2 = 47.88 kN/m2 = 47.88 kPa
1 N/m2 = 20.885 x 10-3 lb/ft2
1lb/ft3 = 0.1572 kN/m3
1 kN/m2 = 1 kPa = 20.885 lb/ft2 = 20.885 x 10-3 kip/ft2
1lb/in3 = 271.43 kN/m3
1 kN/m3 = 6.361 lb/ft3 = 0.003682 lb/in3
CH8-40